OPTIMAL DECISION MAKING FOR SPARE PART MANAGEMENT WITH TIME VARYING DEMANDS. A Thesis by. Venkata Ranga Sri Rama Sunny Repaka

Transcription

1 OPTIMAL DECISION MAKING FOR SPARE PART MANAGEMENT WITH TIME VARYING DEMANDS A Thess by Venkata Ranga Sr Rama Sunny Repaka Bachelor of Engneerng, Andhra Unversty, 2004 Submtted to the Department of Industral and Manufacturng Engneerng and the faculty of the Graduate School of Wchta State Unversty n partal fulfllment of the requrements for the degree of Master of Scence December 2007

2 Copyrght 2007 by Venkata Ranga Sr Rama Sunny Repaka All Rghts Reserved

3 OPTIMAL DECISION MAKING FOR SPARE PART MANAGEMENT WITH TIME VARYING DEMANDS I have examned the fnal copy of ths thess for form and content, and recomm that t be accepted n partal fulfllment of the requrement for the degree of Master of Scence, wth a major n Industral Engneerng. Hatao Lao, Commttee Char We have read ths thess and recomm ts acceptance: Janet Twomey, Commttee Member Lawrence Whtman, Commttee Member Ramazan Asmatulu, Commttee Member

4 DEDICATION To my beloved parents, and my dear frs v

5 ACKNOWLEDGMENTS I would lke to express my sncere thanks and apprecaton to my advsor Dr. Hatao Lao for hs patent gudance, advce, precous tme and constant support throughout my graduate career at Wchta State Unversty, wthout whch my efforts would not have been complete and frutful. I would lke to thank Dr. Janet Twomey, Dr. Lawrence Whtman and Dr. Ramazan Asmatulu for ther precous tme, helpful comments and suggestons throughout the span of ths research. I would lke to thank my frs, Mr. Keerth Dantu, Mr. Dev Kran Kalla, Mr. Zhaojun L, Mr. Subash kodur, Mr. Jogeswara Rao Trlang, Mr. Jayanth Velkur, Mr. Keerth Varala, Mr. Sandeep Vummanen, Mr. Krshna Teja Sangaraju, Mr. Venkat Krshna Srkonda, Mr. Roht Reddy Ramanagar, Mr. Nshanth Mehta, Mr. Chranjeev Mangamur, who encouraged and supported me durng my graduate studes. Fnally, I would lke to thank my famly, for ther moral support and love, as well as ther encouragement and example that motvated me to further my educaton. v

6 ABSTRACT Controllng spare parts nventory of a product s a major effort to many companes. The problem becomes more challengng when the nstalled base of the product changes over tme. In order to cope wth the stuaton, the nventory value needs to be adjusted accordng to the resultng non-statonary mantenance demand. Ths problem s usually encountered when a manufacturer starts sellng a new product and agrees to provde spare parts for mantenance. In ths research, a specal case nvolvng a new non-reparable product wth a sngle spare pool s consdered. It s assumed that the new sales follow some popular stochastc processes, and the product s falure tme follows the Webull dstrbuton or Exponental dstrbuton. The mathematcal model for the resultng mantenance demand s formulated and calculated through smulaton. Based on the mantenance demand, a dynamc ( Q, r) - (lotsze/reorder-pont) restockng polcy s formulated and solved usng a mult-resoluton approach. Fnally, numercal examples wth the objectve of mnmzng the nventory cost under a servce level constrant are provded to demonstrate the proposed methodology n practcal use. v

7 TABLE OF CONTENTS Chapter Page 1. INTRODUCTION Motvaton and Background Overvew of the (Q, r) model Research Objectve 3 2. LITERATURE REVIEW Revew of Sngle-echelon Systems Revew of Mult-echelon Systems Revew of Crtcalty Based Approach Analyss METHOD Modelng of Mantenance Demand Consderng New Sales Mantenance Demand Approxmaton Model Development Soluton Technque NUMERICAL EXAMPLES Expected Number of Stock-Outs Expected Mean and Varance of Stock-outs Falure Rate of the Part Followng Exponental Dstrbuton CONCLUSIONS AND FUTURE WORK...56 REFERENCES..58 APPENDICES...61 A. Matlab Code for Demand Followng Unform Dstrbuton 62 B. Matlab Code for Demand Followng Homogeneous Posson Process 64 C. Matlab Code for Demand Followng Non-Homogeneous Posson Process 67 D. Flowchart of Model Implementaton...71 v

8 LIST OF TABLES Table Page 4.1 Expected Mantenance Demand n a 1 Year Perod for Unform Dstrbuton Optmal Settngs for ( Q, r ) wth Varous Resoluton Levels n Case of Demand Followng Unform Dstrbuton Expected Mantenance Demand n a 1 Year Perod for Homogeneous Posson Process Optmal Settngs for ( Q, r ) wth Varous Resoluton Levels n Case of Demand Followng Homogeneous Posson Process Expected Mantenance Demand n a 1 Year Perod for Non-Homogeneous Posson Process Optmal Settngs for ( Q, r ) wth Varous Resoluton Levels n Case of Demand Followng Non-Homogeneous Posson Process Expected number of stock-outs n a perod of 1 year for dfferent set-ups Expected number of stock-outs for each month n case of m = Mean and Varance of stock-outs for dfferent set-ups Mean and Varance of stock-outs for each m = Mean and Varance of stock-outs for each m = Mean and Varance of stock-outs for each m = Mean and Varance of stock-outs for each m = Expected mantenance demand n a 1 year perod wth falure rate of the part followng exponental dstrbuton Optmal Settngs for ( Q, r ) wth varous resoluton levels wth falure rate of the part followng exponental dstrbuton Mean and Varance of stock-outs for dfferent set-ups n case of exponental dstrbuton.51 v

9 LIST OF TABLES (contnued) Table Page 4.17 Mean and Varance of stock-outs for each m = 1 n case of exponental dstrbuton Mean and Varance of stock-outs for each m = 2 n case of exponental dstrbuton Mean and Varance of stock-outs for each m = 6 n case of exponental dstrbuton Mean and Varance of stock-outs for each m = 12 n case of exponental dstrbuton.55 x

10 LIST OF FIGURES Fgure Page 1.1 Illustraton of (Q, r) Inventory Control Strategy Sngle-echelon System Mult-echelon System Component Structure wth Commonalty Mantenance Demand vs. New sales Aggregate Mantenance Demand Consderng New Sales Mult-resoluton Mantenance Demand Approxmaton Three-nterval (Q, r) Spare Parts Inventory Plot of Expected Mantenance Demand n a 1 Year Perod for Unform Dstrbuton Total Cost vs. m for Unform Dstrbuton Plot of Expected Mantenance Demand n a 1 Year Perod for Homogeneous Posson Process Total Cost vs. m for Homogeneous Posson process Plot of Expected Mantenance Demand n a 1 Year Perod for Non-Homogeneous Posson Process Total Cost vs. m for Non-Homogeneous Posson process Plot of expected number of stock-outs n each case for a perod of 1 year Plot of mean and varance of stock-outs for dfferent set-ups Plot of mean and varance of stock-outs for each m = Plot of mean and varance of stock-outs for each m = Plot of mean and varance of stock-outs for each m = 6 47 x

11 LIST OF FIGURES (contnued) Fgure Page 4.12 Plot of mean and varance of stock-outs for each m = Plot of expected mantenance demand n a 1 year perod wth falure rate of the part followng exponental dstrbuton TC (Total Cost) vs. m n searchng the optmal soluton Plot of mean and varance of stock-outs for dfferent set-ups n case of exponental dstrbuton Plot of mean and varance of stock-outs for each m = 1 n case of exponental dstrbuton Plot of mean and varance of stock-outs for each m = 2 n case of exponental dstrbuton Plot of mean and varance of stock-outs for each m = 6 n case of exponental dstrbuton Plot of mean and varance of stock-outs for each m = 12 n case of exponental dstrbuton Mult-resoluton mantenance demand approxmaton wth unequal tme ntervals...57 x

12 CHAPTER 1 INTRODUCTION 1.1 Motvaton and Background Inventory control s one of the most mportant tasks faced by product manufacturers. In addton to the raw materal, work n process, and fnshed good nventores, spare parts nventores contrbute to large percentage of the overall nventory nvestment. In ths report we lmt our dscusson to the area of spare parts nventores. Compared to other nventory systems, t s more dffcult to control spare parts nventory of a product to meet certan mantenance demand. In most cases, effectve mantenance of spare parts plays a key role n mprovng the manufacturer s reputaton. On the other hand, because a sgnfcant porton of company s total nvestment les n the hands of nventory, t s always mportant to mantan optmal number of spare parts n the nventory. The three most mportant thngs to be satsfed by any nventory system are to provde rght quantty wth rght qualty at the rght tme. Ths research s expected to provde some mportant nsghts that would reduce the spare parts nventory nvestment. There are several thngs to be taken care of whle mantanng optmal number of parts n nventory. We concentrate here on the falure tme dstrbuton of the part whch s a key factor that dentfes the number of parts requred to be mantaned n nventory. The problem we are concentratng on s to nvestgate the unexpected fluctuatons n demand. Some research approaches have been reported n the lterature, ths research s expected to enrch ths area. Spare parts nventory control s usually mplemented to maxmze the avalablty of systems beng served. Snce the amount of tme the machne s down largely deps 1

13 on the avalablty of the part, one has to make sure that the part s avalable when requred. Therefore, the problem les on an effectve balance between the servce offered and cost ncurred. 1.2 Overvew of the ( Q, r) model: Regardng the control of nventory systems, two models are wdely used: the lotsze/reorder-pont (Q, r) model where Q represents the order quantty and r s the reorder pont, and the reorder-pont/order-up-to-level (s, S) where S represents the order-up-tolevel and s s the reorder pont. The model developed n our paper tres to lnk up the assumptons made by the (Q, r) model wth the challenges faced by the company and come up wth an optmal soluton for the problem. The basc dea behnd the (Q, r) model s to determne how much stock to carry and how many to produce or order at a tme. Q -Order quantty I N V E N T O R Y Q θ r l θ - re-order pont - Lead tme -Demand durng lead tme r l Tme Fgure 1.1 Illustraton of ( Q, r) nventory control strategy Assumptons that are made by the ( Q, r) model nclude 2

14 No product nteractons. Unflled demand s backordered. Lead tmes are fxed and known. There s a fxed cost assocated wth each replenshment order. 1.3 Research Objectve The objectve of the research presented n ths report s to mnmze nventory nvestment of a company by mantanng optmal number of spare parts n nventory. Ths research addresses a spare parts nventory control problem for a non-reparable product consderng new sales. When the product fals, t wll be replaced by a new one f spare parts are avalable. A (Q, r) nventory control problem s formulated and the operatng parameters are optmzed based on the resultng mantenance demand. We solve the problem of mantanng optmal number of spare parts usng a mult-resoluton approach that s expected to have a greater mpact on the overall nventory nvestment. Ths report would help personnel n the nventory control department whle dealng wth the ssues such as when to order and how much to order. In order to meet the stated objectve effectve forecastng of the demand for spare parts s requred and t plays a vtal role n movng towards the objectve. As the demand for the spare parts changes over tme, we try to do a perodc revew of the demand and come up wth a seres of decsons regardng the values of optmal order quantty (Q) and reorder pont (r) for each perod based on the falure rate of the parts. Snce we try to do a perodc revew, n addton to the order quantty and reorder pont, the other mportant varable to be determned here s the number of set ups to be made (.e., the number of tmes the values of Q and r have to be changed) durng a certan perod of tme. 3

15 The remander of ths report s organzed as follows. In Chapter 2, we look over the prevous lterature concentratng on the area of spare parts management and gan some mportant nsghts that helped us formulate the optmzaton problem. Chapter 3 addresses the (Q, r) spare part nventory control problem and provdes a mult-resoluton approach to solve the problem. In Chapter 4, numercal examples are provded to demonstrate the proposed approach n practcal use. Fnally, conclusons are gven n Chapter 5. 4

16 CHAPTER 2 LITERATURE REVIEW Snce nventory s a huge topc, there are several papers talkng about effectve mantenance of nventory. For a good revew over the mportance of spare parts management, the reader can refer to the paper [1] by Sandeep Kumar and paper [2] by Kennedy et al. The paper by Kennedy et al. gves a good revew over the recent lterature on spare parts nventores. They dscuss about the future areas that are open for research and explan the condtons that have sgnfcant mpact on mantenance nventory levels. The author dvdes the paper nto dfferent sectons each concentratng on varous problems (lke management ssues, age-based replacement, mult-echelon problems, obsolescence, reparable spare parts) to be solved whle determnng optmal no. of spare parts. The exstng spare parts nventory models can be classfed nto sngle-echelon nventory systems and mult-echelon systems. Fgure 2.1 depcts the behavor of a sngleechelon system and Fgure 2.2 shows a typcal vew of a mult-echelon system wthout any lateral shpments among regonal facltes. Central Warehouse Customer 1 Customer 2 Customer 3 Customer n Fgure 2.1 Sngle-echelon system 5

17 Central Warehouse RF 1 RF 2 RF 2 RF 4 C1 C2 C3 C4 C5 C6 C7 Cn-1 Cn RF: regonal facltes located near to the customers C: customer Fgure 2.2 Mult-echelon system Based on the operatng parameters, these models can be further classfed nto fxed quantty model and fxed perod model. To determne the optmal values of the assocated operatng parameters, mantenance demands: external vs. nternal, determnstc vs. random, statonary vs. non-statonary, need to be characterzed frst, as the optmal soluton s usually senstve to such characterstcs. 2.1 Revew on Sngle-echelon Systems The paper [3] presents an approxmate analytcal model for comparng system approach wth a smpler tem approach. System approach reduces nventory nvestment by showng dfference n fll rates between parts wth hgh costs and parts wth low costs, whle tem approach does not vary fll rates by parts and assgns dentcal fll rates to all knds of parts. The exact and contnuous expressons for these two approaches are developed, to quantfy the percentage reducton n nventory nvestment. In case of exact analyss a contnuous-revew ( S, S ) nventory polcy s used and demand durng the 1 lead tme s assumed to follow a Posson dstrbuton. Also, t s assumed that demand for 6

18 spare parts have the same lead tme. In case of contnuous approxmatons the parameters unt cost and demand rate of the parts are used to represent the skewness of the values across the system. And the demand durng the lead tme s assumed to follow a Laplace dstrbuton. Fnally, they compare the values of nventory nvestment under Posson demand and compound Posson demand and numercal results show that relatve mprovement n nventory nvestment deps largely on the skewness of the parameters. The paper [4] evaluates the performance of two dfferent (Q, r) models; a smple and an advanced model. The smple model assumes that demand durng lead tme follows normal dstrbuton and the advanced model assumes that demand follows gamma dstrbuton. The paper nvestgates the two models usng Monte Carlo smulaton and shows that the advanced model consstently outperforms the smple model and that the servce levels obtaned are close to the desred servce levels usng the advanced approach. The output of the smulaton program s gven by P 2 = 1 (mean shortage per cycle/mean demand per cycle) [4], where P 2 s the attaned servce level. It also gves the average nventory level on the bass of smulaton and accordng to the formulae generated. Also, the mportant nsght of ths paper s that t shows the sgnfcant effect of the varance of the forecast error durng lead tme when compared to the varance of the demand durng lead tme. Paper [5] consders a system where varous parts are mantaned to support the repars for a set of products. The objectve of ther work s to determne an optmal stockng polcy for each part whle mnmzng the expected holdng and orderng costs and meetng the servce constrants. They develop a heurstc procedure based on the dualty theory and show that t s closely related to the well known Greedy heurstc. 7

19 They use a perodc revew base stock polcy and all the parts are stocked to the level S at the begnnng of each perod (replenshment occurs nstantaneously). For the model developed, two forms of servce level constrants (chance constrants used by Mtchell (1982, 1983) and part avalablty constrants used by Beesack (1967)) are consdered. In extenson to the basc model whch does not take nto account any prorty customers or parts commonalty among products (see Fgure 2.3); they ext ther research to the case where a low and a hgh prorty customer class and also products havng smlar parts are consdered. Product Famly Product Product j Part 1 Part k Part n Part p Fgure 2.3 Component structure wth commonalty [5] In addton to the prce and qualty of the parts, companes often try to reduce the cycle tme n order to gan a compettve advantage. Schultz [6] n hs paper tres to address ths ssue by quantfyng the effect of spare parts nventory on the processng tme of the machnes. They present an exstng closed-form approxmaton to llustrate how the mean and varance of machne repar or downtme affects average cycle tme and departure varablty at a gven workstaton and downstream workstatons. They ntally 8

20 consder the case of a sngle-crtcal component and then ext ther research to the case where there are multple-crtcal components n a sngle machne. The model developed assumes that components fal exponentally and t ams at mnmzng the expected backorders subject to an nventory nvestment constrant. Also, t s assumed that nventory s controlled by a one-for-one replenshment polcy and repar tme s taken as neglgble f a spare part s avalable. The mean tme to repar or downtme of a machne largely deps on the avalablty of the spare part, so here the author tres to maxmze the avalablty of machne whle mnmzng the costs assocated wth nventory. By takng some numercal examples the paper shows how the annual proft can be ncreased by mantanng optmal number of spare parts. In process of effectve mantenance of spare parts, the frst and foremost step s to estmate the demand wthn a certan perod of tme. In the paper [7] they present a case study for applyng Bayesan approach to forecast demand and to come up wth an optmal parameter S for an ( S 1, S) nventory system that stocks spare parts for electronc equpment. They consder the case of a company that produces crcut packs as spare parts whch are very much essental for contnuous runnng of the electronc equpment. As the demand for spare parts occurs by the random falure of the nstalled parts, the polcy suggested here ams at decreasng the base stock level by a sgnfcant amount. They ntally present the concepts of Bayesan approach and then show the results of applyng Bayesan approach to forecast demand when compared to the current polcy used by the frm. The current polcy operates on the assumpton that tme to falure of the parts follow exponental dstrbuton and t calculates the value of 'S' for a desred servce 9

21 level p. Equaton 2.1 gves the formula for the calculaton of S whle satsfyng the servce level of the customer. S k = 0 1 ( nl) k e nl / k! p. 2.1 Here L s the lead tme and n s the number of crcut packs nstalled. The equaton used above closely resembles to the equaton that s used n our report for the calculaton of reorder pont n our optmzaton problem. As the current polcy takes the parameters as known and constant, the author consders the case where a new part s ntroduced nto the market and no real data s avalable over the falure rate of the part. In order to account for ths, the author assumes a compound Gamma-Posson probablty functon for the number of falures k durng the replenshment lead tme L and calculates the crtcal stock level that satsfes the desred servce level. The proposed method sgnfcantly lowers the base stock level whle achevng the desred servce level. The paper [8] concerns about provdng a general and smple algorthm whle obtanng an optmal soluton for dfferent nventory models n a manufacturng system wth a lnearly ncreasng or decreasng tr n demand. Ther man focus was to come up wth a replenshment or producton schedule that mnmzes the total cost of the system. The model developed n ths paper acts as a general equaton and reduces the complexty nvolved n solvng the problem. They conduct the study under a fnte tme horzon and theoretcally prove that total cost s a convex functon of the number of producton cycles or replenshments done by the company. However, the results obtaned from the above approach just determne the optmal number of replenshments to be made and does not talk about other operatng parameters that control the stock mantaned wthn the nventory. The model presented n ths report presents optmal 10

22 operatng parameters for each perod whle controllng the non-statonary mantenance demand and satsfyng the servce level specfed to the customer. Spare parts are often kept n stock n order to support mantenance operatons and to protect aganst equpment falures. The paper [9] conducts a case study at a large ol refnery for comparng dfferent re-order pont polces that optmze the number of parts mantaned n nventory. However, t s always dffcult to effectvely manage spare parts that are very slow movng wth hghly stochastc and erratc demands. In order to reduce the dffculty of assessng lead tme demand dstrbutons for these knds of parts, the paper [9] tres to test dfferent theoretcal models wth real demand data taken from a company. The paper studes the case where a company holds 43,000 catalogued materals wth a total cost of more than 27 mllon euros. As controllng a huge number of parts always represent a dffcult task due to the varatons n crtcalty and demand for the parts, the company mantans stock by movng to the SAP R/3 nformaton system that operates n terms of mn-max levels where the mnmum level s s determned wth the help of safety stocks and maxmum level S determned by lot szng methods. As the prce, demand and crtcalty of the part vares, the paper defnes a combned class xyz (where x represents demand class, y represents crtcalty class and z represents prce class) for each tem and optmzes the operatng parameters for each ndvdual class n process of dentfyng the best optmzaton rule for the system. Two types of approaches (ex-ante and ex-post) are developed for optmzaton of the spare parts. In case of ex-post method whole data set s used for both fttng and testng purposes, whereas n case of ex-ante approach the hstorcal data that s avalable s dvded nto two dfferent sets where one s used for fttng the dstrbuton and the other 11

23 for performng a smulaton and comparng wth the performance of the current one. After obtanng the optmal parameters for each dstrbuton and calculatng the total cost of the system, the author compares each nventory model wth the current system and shows that of all the nventory models consdered normal model performs really well achevng total savngs of 8.4% (about 1.3 mllon euros) over the current system. The paper [10] deals wth the case of a frm that offers servce contracts on ts equpment and stocks a huge varety of parts to support feld mantenance of ts equpment. Ther man objectve s to reduce the nventory nvestment by meetng the servce level and order frequency constrants that are set by the user. As the author here concentrates on the mantenance of spare parts that are requred n order to ncrease the avalablty of the equpment, n our report we try to do t by concentratng on the falure rate (age) of the parts whch defntely have a sgnfcant mpact on the overall nventory nvestment. Here they present three heurstc algorthms whch are easly mplementable and nsenstve to the addton or subtracton of new parts. They compare the three heurstcs developed wth the methods prevously n use by the frm and show that these methods are more effcent and same customer servce level can be attaned at 20-25% smaller nventory nvestment. In process of developng the model, they make some assumptons that nclude full backorderng; demand follows a Posson process; lead tmes are fxed; demand occurs n unts of batches, batch sze s constant and steady state model rrespectve of the age of the parts. In ths report we relax the assumptons that demand follows a Posson process and a steady state model for the parts and take that product s falure tme follows Webull dstrbuton. The author tres to solve the problem usng the well known ( Q, r) model and 12

24 come up wth a good soluton. In order to make t smpler he frst developed the model for a sngle product case and then exted ther work to a mult product case. The constraned optmzaton model for the sngle product case s shown n equaton 2.2. Mnmze c(r - θ + Q/2) 2.2 Subject to λ/q F, 1 - A(r, Q) S, r r-, Q 1, r, Q : ntegers where c s the unt cost of the tem, and (r - θ + Q/2) s the on hand average nventory at any pont of tme assumng that safety stock s always greater than or equal to zero.in detalθ s the expected lead tme demand, λ (mean demand per year), F and S are target order frequency and servce level specfed by the user, A(r, Q) s the probablty of stock out at any pont of tme. After the development of heurstcs, by takng a numercal example, they compare the effectveness of heurstcs and conclude that type 1 heurstc whch calculates the values of order quantty and reorder pont works well when the servce level s hgh, hybrd heurstc whch s a mxture of type 1 heurstc and type 2 heurstc works well when the order frequency s hgh and type 2 heurstc whch requres both the order quantty and reorder pont to be computed consecutvely works better than the above two heurstcs. 2.2 Revew on Mult-echelon Systems Servce agreements made by supplers to the customers play a vtal role n mantenance of spare parts n any ndustry. There are several papers talkng about ths ssue. The paper [11] develops a contnuous-revew nventory model for a mult-tem, 13

25 mult-echelon spare parts dstrbuton system wth tme-based servce level requrements. The objectve of ths paper s to determne base stock levels for all parts at each locaton whle mnmzng the nventory nvestment. The model developed n ths paper ncludes multple constrants wth dfferent tme settngs across random combnatons of tems and demand locatons. The paper derves exact and contnuous expressons for each tem and ntally develops a greedy algorthm to fnd near optmal solutons and then t develops a prmaldual procedure (Lagrangan based approach) to fnd near optmal solutons as well as lower bounds at all locatons. The author fnally compares and shows the effectveness and scalablty of these dfferent procedures by takng three example problems. The paper [12] n extenson to ther prevous work, consdered a two-echelon nventory system that has a dstrbuton centre (DC) and regonal facltes that stocks varous spare parts requred for the system. As the outages due to falures are not supposed to exceed certan number of hours per month, n exchange for the servce level constrant that s prevously n use by the frm, here they consder an optmzaton problem where the average total delay due to falures s to be mantaned below a specfed level. In support for the development of the model, they assume that the DC operates on the bass of contnuous revew (Q, r) polcy and the facltes use the base stock polces for the replenshment of parts from the DC. The model formulated n ths problem can be expressed as follows: N Mnmze c h Subject to: ( - total nventory nvestment 2.3 M 0 r0, Q0 ) + hm ( S m ) = 1 m= 1 1 λ F; N 0 N = 1 Q0 14

26 N B m ( S = 1 m ) T m, m = 1,.., M. Here N = number of tems, c = cost of tem. h r, Q ) = expected on hand nventory of tem at DC at any pont of 0 ( 0 0 tme h ( S ) = expected on hand nventory of tem at faclty m at any pont m m of tme λ 0 = expected annual demand for tem at the DC (parts/yr) Q 0 = order quantty for tem at at the DC F = target order frequency B ( S ) = expected number of backorders for tems at faclty m at any m m pont of tme T m = total delay that s allowed per year at faclty m As the model developed represents a large scale non lnear dscrete optmzaton problem, they decompose the problem and develop closed form expressons for the control parameters at the faclty and the DC. The paper [13] consders a smlar system that has a sngle central warehouse and several local warehouses to support the falure of the parts occurrng at varous customer locatons. Here they dvde the customers nto several classes and assgn a target aggregate fll rate to each class. They develop a mult-tem, sngle stage spare parts nventory system for ths problem. It s assumed that customers have smlar machnes and falure rate of the part follows a Posson process. A crtcal level s specfed for all the tems per customer class. The satsfacton of demand for part from a customer class j 15

27 deps on ths crtcal level, the hgher the crtcal level, the lower the servce provded for the customer. The optmzaton problem developed for ths case s gven by: I Mn = 1 p c, (nventory nvestment) 2.4 J + 1 I m, j Subject to β ( c ),,, j β j obj M = 1 j (fll rate constrant) 0 = c, 1... c, J c, J + 1, [13] Here p denotes the prce of the tem and c, j+ 1 represents the crtcal level for customer j + 1. As the problem s a nonlnear nteger optmzaton problem, they develop a heurstc soluton procedure based on Lagrange relaxaton to obtan a good approxmate soluton that generates both heurstc soluton and lower bound for the optmal costs. After the development of the model, expressons for the tem fll rates are derved n order to evaluate the effectveness of the crtcal level polcy followed. Also, the parameters (Lagrange multplers) are optmzed to get the best feasble soluton. Computatonal results show that the error between the costs of lower bound and heurstc soluton decreases wth the ncrease n number of tems and applyng crtcal level polces reduces the nventory nvestment n comparson to the general base stock polces wth equal aggregate fll rate to all the customers. The paper [14] deals wth mantanng optmal number of spare parts n a multlevel mantenance system. The paper develops computer programs (MASOPS a FORTRAN program) for calculatng the optmal number of spares at two and three mantenance levels. They consder a herarchcal, pyramdal system that has three supply levels of nfnte capacty and complete dversty. The optmzaton procedure developed 16

28 ams at mnmzng the costs of spare part kts whle maxmzng the mantenance effcency. Spare part kts are assgned for each mantenance level and spare parts at lower level are replenshed from spare parts at hgher level. The paper classfes all the tems nto dfferent groups and develops an algorthm for calculatng the optmum spare part kts. Later, wth the help of a numercal example they llustrate the mpact of mantenance polcy on the effcency of the mantenance system. The paper [15] s about effectve mantenance of a two-echelon spare parts nventory system. They consder a system that controls equpment falure at varous locatons wth the help of a central warehouse and several feld depots (that stocks spare parts) placed at nearest locatons to the customer. They develop a contnuous revew, base stock polcy to mnmze the overall nventory cost subject to a servce constrant at each feld depot. The model developed n ths paper uses response tme of the suppler as the servce constrant and ams at keepng t below a threshold level of 4 hours. In process of determnng nventory polces at the central warehouse and feld depots, they ntally develop expressons for fndng nventory and backorder levels at all the depots and then present a heurstc algorthm that can be solved for obtanng best soluton and best lower bound for the base stock levels at ndvdual depots as well as warehouse. They also show the effectveness of the algorthm when compared to the heurstc developed n the paper [12]. 2.3 Revew on Crtcalty Based Approach As the management of spares has a bg role to play n maxmzng the profts for any company, Wang et al. [16] try to optmze the spare part stores by usng an mproved genetc algorthm n a supply chan knd of system. In process of managng the spare 17

29 parts, they classfy them nto ABC classes based on the captal proporton and usage lfe of the parts. The author descrbes some of the key ssues (lke checkng and evalu tng the status of equpment, settng up a management nformaton system) that keep update of the requrement of spare parts n tme. Based upon the mproved genetc algorthm they develop a two-objectve optmzaton model that mnmzes the captal cost and maxmzes the safety factor. In the smulaton phase, the model developed shows that 39% of the overall cost can be saved when compared to the tradtonal method followed. Yang et al. [17] proposed four crtcalty evaluaton methods that control ntal provsonng for spare parts. Falure of the crtcal parts often leads to the total shut down of the system. Here, based on the crtcalty of the parts they estmate the number of parts to be mantaned that decreases the total lfe cycle cost of the system. The four methods proposed n ths paper nclude the RPN (rsk prorty number), the GRN (grey relaton number, the product of RPN and MTAT (mantenance turn around tme), and the GRN wth MTAT. By consderng the RPN and MTAT, the paper [17] determnes the crtcalty of the tem and assgns weght for the tem. In order to llustrate the methods, the paper studes the case of an gnton system that has several components wth varyng MTBF, falure rate and unt cost. The optmal spares stockng polcy among the four methods s selected based upon the cost effectveness curves of the system. 2.4 Analyss From the above lterature t can be noted that most research has been focused on ether (Q, r) or (s, S) nventory control problems assumng statonary mantenance demands,.e., ther dstrbutons do not change over tme. The dstrbutons are usually determned by fttng hstorcal demand data or are smply assumed based on experence. 18

30 However, ths approach s not approprate when new sales are consdered, whch expand the nstalled base over tme. In fact, under such crcumstance a non-statonary stochastc process s more approprate for modelng the resultng mantenance demand, and t s desrable to nvestgate how the product s adopted n order to proactvely forecast the upcomng mantenance demand for better spare parts management. Ths s an mportant factor that ether ncreases or decreases the number of parts requred to be mantaned n nventory at any specfc pont of tme. Ths report studes the case of a new product where the manufacturer agrees to provde spare parts for mantenance. As mantenance demand vares over tme, we do a perodc revew of the demand and try to reduce the overall nventory nvestment n a sngle-echelon system whle satsfyng the customer. 19

31 CHAPTER 3 METHOD In ths chapter, we propose an nventory model that ams at mnmzng the nventory nvestment of a company whle meetng the servce level specfed to the customer. In order to succeed n today s compettve market t became mportant for companes to see that they delver ther parts at rght tme wth rght qualty. Therefore companes often try to ncrease the avalablty of the system whch s possble through effectve mantenance of spare parts n nventory. Provdng the spare parts when requred ncrease the up-tme of the systems, whch s crucal for on-tme delvery of the products. As ncreasng or decreasng the number of spare parts mantaned hghly affects the profts made by the company, they have an mportant role to play n reducng the operatonal costs of the company. Here we try to reduce the nventory nvestment by mantanng optmal number of spare parts n nventory. In order to make ths happen t s requred to effectvely estmate the number of spare parts requred at any specfc pont of tme. The number of spare parts requred largely deps on the falure rate of the parts. Effectve control of spare parts s a tough task faced by the personnel n the nventory management department today. Systems often start to deterorate as the age and usage of the system ncreases. As the falure rate of parts ncreases wth tme, the demand for spare parts also ncreases n proporton wth tme. Studyng the falure rate of the parts helps one to effectvely predct the future performance of the system. In regard to these ssues as there are dfferent types of techncal systems that requre spare parts, we consder the case of a new non-reparable product that s servce contracted to the 20

32 customer [18]. And as the orgnal manufacturer of the product s responsble for provdng the spare parts requred to maxmze the avalablty of the system, we here come up wth a model that deals wth ths ssue whle mnmzng the mantenance cost of the system. Also as the orders from the clents come wthout any pror ntmaton, the nventory control department feels dffcult to satsfy the customer when the order s placed for slow movng parts. In ths report we ntally assume that rate of new sales of the product s constant and the falure tme of the product follows Webull dstrbuton. In process of solvng the model we consder a fnte perod of 12 months wth demand for spare parts based on the falure rate of the parts. We are just concerned about one sngle component and we do not talk about the ssues such as whch tems to be treated as spare parts and whch tems to be kept n stock? [18]. The model developed s for a sngle echelon system where the parts requred are shpped drectly from the central warehouse to the customer wthout any lateral transshpment between the warehouses. The crtcal component for whch the optmal stock s determned s consdered to be a non-reparable part, and ts state s as good as new after each nstant replacement. Spare parts are often kept n stock n order to control the lead tmes that t takes to replace the defectve part. As the requrement for spare parts deps on the preventve mantenance actvtes followed by the clents, we assume that the company makes contract wth all the clents that certan rules were followed for performng mantenance actvtes n order to ncrease the accuracy of the demand forecasted. 21

33 3.1 Modelng of Mantenance Demand Consderng New Sales [19] Before formulatng the mantenance demand, the followng assumptons are made: 1. At the tme t = 0, the frst unt s adopted by the customer. New sales occur one at a tme followng the homogeneous Posson process wth the rate of λ, the probablty mass functon s gven by: λt n e ( λt) Pr( N( t) = n) = (3.1) n! 2. The product s non-reparable, and ts falure tme follows the two-parameter Webull dstrbuton wth pdf: β t f ( t) = η η t exp η β 1 β (3.2) where β > 0 and η > 0 are the shape parameter and scale parameter, respectvely. 3. Compared to the total lfe tme of the product, the total servce tme (e.g. replacement tme and tme spent watng for a spare part f any) are gnored (.e. nstant replacement) It can be noted that, as the product s non-reparable and as good as new after each nstant replacement, each unt of product adopted wll generate a renewal process. Let F (t), t > 0, be the cumulatve dstrbuton functon (Cdf) of the product s falure tme, and H (t), t > 0, be the correspondng renewal functon (expected number of renewals). Accordng to the renewal theory, H (t) can be expressed as: 22

34 t ( ) H (t) = F (t) + H ( t s) df( s) = F ( t), 0 = 1 (3.3) Ths equaton gves the expected mantenance demand requested by a sngle unt ( ) nstalled at tme 0. In the above equaton F ( t) represents the - fold convoluton of the underlyng falure tme dstrbuton F (t), whch s defned as F ( ) ( t) = F t 0 ( 1) ( t u) df( u), t u 0, n 2 (3.4) ( ) wth F ( t) = F( t), Note that, for the two-parameter Webull dstrbuton β t F( t) = 1 exp. For the two-parameter Webull dstrbuton, the analytcal η soluton of ths equaton does not exst, and the equaton has to be solved numercally. In practce, the drect Remann-Steltjes ntegraton approach proposed by Xe [20] may be utlzed. Furthermore, let S (t) be the aggregated expected mantenance demand at tme t. If the product s nstalled base has multple unts (say n ) at tme 0 and there s no new sale afterwards, S (t) can be smply expressed as: S (t) = nh (t), as each product generates ts own renewal process. When new sales are consdered, each unt newly adopted wll start ts renewal process from the tme when t s adopted. Fgure 3.1 llustrates the mantenance demand under such crcumstance. As a result, S (t) becomes the aggregaton of the expected number of renewals of multple renewal processes startng from dfferent ponts n tme. The mantenance demand for the case where new sales follow homogeneous Posson process can be derved as follows: 23

35 Fgure 3.1 Mantenance demand vs. new sales Case 1: When there s no new sale up to tme t (n = 0 wth the probablty of e λt ), the mantenance demand process s same as the renewal process of the frst unt adopted,.e.: S ) Case = H (t). (t { 1} Case 2: When there s only one new sale up to tme t (n = 1 wth the probablty ofλte λt ), the condtonal expected aggregate mantenance demand s: S ) (t { 2} t λt1 Case = H ( t) + H ( t t1) λ e dt1. 0 Case 3: When there are two new sales up to tme t (n = 2 wth the probablty 2 λt of( λt) e / 2! ), the condtonal expected aggregate mantenance demand becomes: S( ) t { 3} t t t 2 1 Case = H ( t) [ H ( t t1) H ( t t1 t 2 )] e λ t λt + λ λe dt2dt Case n: When there are new sales up to tme t (n = wth the probablty λt of( λt) e /! ), the condtonal expected aggregate mantenance demand becomes: 24

36 S (t) { Casen } = H ( t) + t t t1 t t1 t2 t t1 t2... t H ( t j= 1 k = 1 j tk ) λ e λ t m m= 1 dt dt 1... dt. In summary, consderng all the possbltes the expected aggregate mantenance demand at tme t can be expressed as: S (t) = H (t) + t t t1 t t1 t2 t t1 t2... t 1 j... H ( t t = j= 1 k= 1 tm m= 1 k ) λ e dt dt 1... dt 1 e λ λt 1 ( λt)! (3.5) Note that the expected number of mantenance demand durng the tme nterval [ t, t + t) s gven by: S ( t + t) - S (t). Equaton (3.5) gves the exact formulaton of the aggregate expected mantenance demand; however t s dffcult, f not mpossble, to calculate numercally. To overcome ths dffculty, a smulaton based approach s utlzed n ths report. The smulaton program s developed usng Matlab, whch generates the tme of each new sale as well as correspondng falure tmes after the sale. Whle generatng demand usng Matlab we assume that the falure tmes of the parts follow Webull dstrbuton wth scale parameter ( η = 1) and shape parameter ( β = 2 ). Fgure 3.2 shows the aggregate mantenance demand (from 100 runs) generated by the smulaton program. To mplement ths smulaton approach n estmatng the aggregate expected mantenance demand S (t), the program wll be executed for a large number of runs and S (t) wll be estmated by takng the average from those smulaton runs: Ŝ (t) = S ( t) M (3.6) 25

37 Where S (t) s the aggregate mantenance demand obtaned n the th smulaton run and M s the total number of runs. After generatng the demand we start solvng the model to mnmze the costs ncurred by the company. Fgure 3.2 Aggregate mantenance demand consderng new sales The total demand for each month can be determned by the summaton of demand for new sales and the demand for mantenance. Total Demand = demand for new sales + demand for mantenance. 3.2 Mantenance Demand Approxmaton [19] To control the spare parts nventory, one approach s to update the mantenance demand and adjust the spare parts nventory strategy,.e., the values of ( Q, r), once a new sale occurs. However, ths approach may not be vable n most real world applcatons because of hgh cost ncurred by frequent setups of the assocated operatng parameters. To facltate mplementaton, a mult-resoluton method s proposed, whch dvdes the tme perod beng consdered nto several smaller tme ntervals, n whch the average aggregate expected mantenance demands are calculated by: Ŝ = Sˆ( b ) S ˆ( a ) b a (3.7) 26

38 where a and b are the lower and upper bounds of the th tme nterval, wth S ˆ( ) and S ˆ( ) estmated from equaton (3.6), respectvely. In partcular, wthn each a b nterval the mantenance demand s assumed to follow a homogeneous Posson process. The resoluton of ths approxmaton technque deps on the lengths of those tme ntervals utlzed. Fgure 3.3 shows an example wth a lower resoluton and another wth a hgher resoluton. When smaller tme ntervals (hgher resoluton) are utlzed, the approxmaton to the mantenance demand wll be more accurate. The extreme soluton s to predct the mantenance demand whenever a new sale occurs, but ts economc performance may become worse due to frequent setups, as we mentoned earler. In most ndustry applcatons, the trade off between the approxmaton accuracy and the economc effcency can be made n such spare parts nventory control problem. Fgure 3.3 Mult-resoluton mantenance demand approxmaton 3.3 Model Development [19] To facltate the presentaton of the proposed nventory control model, the followng notatons are ntroduced frst. Notaton M max = the maxmum number of setups allowed 27

39 m = number of tme ntervals to be consdered ( M max ) c h = holdng cost per tem per tme unt c s = shortage cost per tem c o = cost per order c sp = cost of settng up the nventory operatng parameters for each tme nterval L = length of tme nterval c = c L ] [ h h l X = replenshment lead tme = random demand durng replenshment lead tme durng tme nterval θ = expected aggregate demand durng replenshment lead tme n tme nterval r = reorder pont for tme nterval Q = order quantty for tme nterval TC = total nventory cost over the nterested tme horzon The two decson varables that hghly nfluence the stock mantaned n any company are the values of Q (order quantty) how much to order and r (reorder pont) when to order. Optmzng these two values sgnfcantly ncreases the profts made by a company. In general reorderng has been controlled by systems such as MRP, JIT and statstcal nventory control methods that suts the condtons effectvely [16]. Hgher values of Q and r reduce the orderng cost and penalty cost of the company whle ncreases the holdng cost of the company. On the flp sde, lower values of Q and r reduce the holdng cost and ncrease the orderng cost and penalty costs wthn the company. Therefore a perfect trade off has to be provded n order to mnmze the total costs ncurred by the company. 28

40 In our model, we concentrate on these two parameters and try to come up wth a seres of decsons regardng the values of Q and r whle satsfyng the constrants of the company. The optmal values for Q and r are determned based on the predcted demand for parts durng a certan perod. As t s always not possble to accurately forecast the demand, n our model we try to reduce the error caused due to ths by mnmzng the length of the perod for whch forecasts are made. Hence, the model n ths paper comes up wth optmal values of Q and r for each ndvdual perod tryng to mnmze the operatonal costs of the company. The values of Q and r obtaned ncreases as demand s estmated for each perod. By mantanng lower values of Q and r n the earler stage of lfe cycle of the equpment, we decrease the holdng cost of the company whch contrbutes to a sgnfcant part of the captal nvestment made by the company. The nventory system n our model operates on the bass of a contnuous revew polcy wth perodc revew of demand at certan ntervals that helps to revse the values of Q and r whle mnmzng the total cost of the system. Fgure 3.4 shows a good vew of the approach followed for our model. The entre schedule perod s dvded nto three tme ntervals, each of whch has a set of operatng parameters ( Q r ) to be optmzed., Fgure 3.4 Three-nterval ( Q, r) spare parts nventory model 29

41 From Fgure 3.4 we can see that n the ntal stages, an order of quantty Q1 s placed at reorder pont r1 and as the tme ncreases the values of Q and r gradually ncreased. The proposed mult-resoluton spare parts nventory control strategy jontly determnes the number of ntervals m and all the resultng operatng parameters ( Q r ), [ 1, m], n each nterval under gven constrants. In partcular for a gven number of ntervals m, the optmal values of ( Q r ) are determned based on the assocated average, aggregate expected mantenance demands durng those tme ntervals. In ths report, equal-length tme ntervals are utlzed for each resoluton strategy, and the average aggregate expected mantenance demand rate n each tme nterval s calculated usng equaton (3.7). The total descrpton of the method followed can also be seen from the flowchart pctured n Appx C. Also from our spare parts nventory control model, we can see that we are nothng but lowerng the values of Q and r n the ntal stages and expectng to decrease the total cost ncurred by the system. Ths way of decreasng the values of Q and r reduces the holdng cost but ncreases the orderng cost of the system. Therefore, n our model we deal wth the mnmzaton of holdng cost, orderng cost and shortage cost whle meetng the servce level specfed to the customer. Specally, we calculate the shortage cost from the expected number of shortages as follows. The expected number of shortages per cycle can be approxmately evaluated by: E[ number of shortages/cycle] = ( ), 1! x e θ θ x r x= r + x, 30

42 where ˆ θ = l S s the aggregate expected mantenance demand durng the replenshment lead tme nterval. After several algebrac manpulatons, the assocated expected total shortage cost can be obtaned as: r θ x E[ shortage cost/cycle] = c e θ s ( ) θ r x r, 0! x= x where c s s the shortage cost per tem. Let TC be the total nventory cost over the tme horzon nterested. Consderng all cost elements, the mult-resoluton optmzaton model for controllng the spare parts nventory can be formulated as Problem P 1 : mn m {, } Subject to, Q r TC = E[ total nventory cost] n r x θ Q e D = ch r c θ θ + + s r x r + co m csp x x Q θ ( ) + + = 1 2 = 0! m M max r x = 0 e θ θ x! x Q > r Q r ; 0,, Z, for. The model operates under the constrant that the servce level provded by the frm s at least 95% (.e. probablty that the demand s met from the stock mantaned). The frst part of the objectve functon calculates the average nventory mantaned for each perod whch when multpled by the holdng cost of the tem gves the average nventory nvestment for that partcular perod. The second part of the objectve functon gves the penalty cost assocated wth the nventory level mantaned. We calculate ths shortage cost by estmatng the expected number of stockouts for that specfc perod. Snce we are reducng the level of nventory mantaned we are nothng but ncreasng the 31